1
core
Sequences
Real & complex analysis
Study the convergence of \( 1 - \frac{1}{2^2} + \frac{1}{3^3} - \frac{1}{4^4} + \dots \)
infinite-seriesseries-convergenceabsolute-convergencealternating-seriesdirect-comparison-testgeometric-seriescalculus-iievaluate-convergence
Sign in to see the solution and its full explanation.
Convergence of an Alternating Series with n^n in the Denominator →
2
warmup
Antiderivatives
Calculus
Find \( \int \frac{1}{\sqrt{x-1}} \, dx \).
indefinite-integralsubstitutionpower-rulesquare-rootcalculus-ievaluate-integralantiderivativefractional-exponent
Sign in to see the solution and its full explanation.
Integral of 1 over the square root of x-1 →
3
hard
Differential equations
Calculus
Find
\[
\mathcal{L}^{-1}\left(\frac{8}{s+5} + \frac{7}{s^2+3}\right).
\]
laplace-transforminverse-laplace-transformlinearityexponential-functionsine-functionordinary-differential-equationscalculusevaluate-inverse-transform
Sign in to see the solution and its full explanation.
Inverse Laplace Transform of a Rational Function Sum →
4
warmup
Limits
Calculus
\(\lim_{x\to 0} \frac{1 - \cos x}{x}\)
limitl-hopitaltrigonometric-limitindeterminate-formcalculus-ievaluate-limitcosinesinederivative-of-cosinezero-over-zerolimit-evaluationbasic-limit
Sign in to see the solution and its full explanation.
Limit of (1 - cos x)/x as x approaches 0 →
5
warmup
Polar coordinates
Calculus
\(a_{n}=\frac{4n+5}{n^{3}-2n+3}\)
limit-of-sequencerational-functionasymptotic-behaviorcalculus-iisequenceslimits-at-infinitydivide-by-highest-powerevaluate-limit
Sign in to see the solution and its full explanation.
Limit of a Rational Sequence →
6
core
Techniques of integration
Calculus
\(\int x^2 \tan^{-1} x \, dx\)
integration-by-partsinverse-trigonometric-functionspolynomial-long-divisionrational-functionssubstitutionindefinite-integralcalculus-ievaluate-integralnatural-logarithmalgebraic-manipulation
Sign in to see the solution and its full explanation.
Integral of x^2 arctan(x) →
7
warmup
Logarithmic and exponential functions
Calculus
\( \ln \frac{1}{5} \)
logarithmsnatural-logarithmproperties-of-logarithmssimplificationpower-rule-for-logarithmsquotient-rule-for-logarithmstranscendental-functionsprecalculusalgebraexponent-rulesnegative-exponentsevaluate-expression
Sign in to see the solution and its full explanation.
Simplifying the Natural Logarithm of a Unit Fraction →
8
warmup
Trigonometric and inverse trigonometric functions
Calculus
Calculate \( \frac{d}{dx}(\cos x) \) from the known derivative of \( \sin x \).
derivativestrigonometric-functionschain-rulecofunction-identitiessinecosinecalculus-ievaluate-derivativeproof-technique
Sign in to see the solution and its full explanation.
Derivative of Cosine via Sine →
9
hard
Inner product spaces and orthogonality
Linear algebra
Consider vectors \( u = (1,3, -6, 4) \) and \( v = (3, -5,1, -2) \) in \( \mathbb{R}^4 \). Find
(a) \( \|u\|_\infty \) and \( \|v\|_\infty \),
(b) \( \|u\|_1 \) and \( \|v\|_1 \),
(c) \( \|u\|_2 \) and \( \|v\|_2 \),
(d) \( d_\infty(u, v), d_1(u, v), d_2(u, v) \).
linear-algebravector-spacesnormsvector-normsinfinity-normone-normtwo-normeuclidean-normdistancemetricreal-vectorsevaluate-norms
Sign in to see the solution and its full explanation.
Computing the 1, 2, and Infinity Norms and Distances in R^4 →
10
core
Vector spaces
Linear algebra
Simplify:
(a) \(i^0, i^3, i^4\)
(b) \(i^5, i^6, i^7, i^8\)
(c) \(i^{39}, i^{174}, i^{252}, i^{317}\)
complex-numbersimaginary-unitpowers-of-iexponent-rulesdivision-algorithmmodular-arithmeticsimplificationalgebraprecalculus
Sign in to see the solution and its full explanation.
Simplifying Powers of the Imaginary Unit →